Spatial+: A novel approach to spatial confounding

Abstract

In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered modeling forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. We propose a novel approach, spatial+, for dealing with spatial confounding when the covariate of interest is spatially dependent but not fully determined by spatial location. Using a thin plate spline model formulation we see that, in this case, the bias in covariate effect estimates is a direct result of spatial smoothing. Spatial+ reduces the sensitivity of the estimates to smoothing by replacing the covariates by their residuals after spatial dependence has been regressed away. Through asymptotic analysis we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straightforward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non-Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.

Keywords: bias reduction; collinearity; forestry; partial thin plate spline regression; spatial confounding.

FIGURE 1

Forestry example. Estimated effect of minimum temperature in May on crown defoliation in the null model (left) and the spatial model (middle), where for each model the plot shows the contribution of the centered covariate to the predicted response (with two times standard error bands). Estimated spatial effect in the spatial model (right) with the border of Baden‐Württemberg outlined and dots showing the data locations

FIGURE 2

Estimated covariate effect $\widehat{\beta }$ (top) and MSE of fitted values (bottom) for each model fitted to 100 data replicates, where the true covariate effect is $\beta =3$. Sp and Sp+ denote the spatial and spatial+ models, respectively, and superscript 0 refers to an unsmoothed model (ie, $\lambda ={\lambda }_x=0$). Results in gray are the three models that correspond to those used in Thaden and Kneib's simulation study. This figure appears in color in the electronic version of this article, and any mention of color refers to that version

FIGURE 3

For each of the distributions Poisson (top), exponential (middle), binomial (bottom): the estimated covariate effect $\widehat{\beta }$ (left) and log(MSE) of fitted values (right) for each model fitted to 100 data replicates, where the true covariate effect is $\beta =3$. Sp and Sp+ denote the spatial and spatial+ models, respectively, and superscript 0 refers to an unsmoothed model (ie, $\lambda ={\lambda }_x=0$). This figure appears in color in the electronic version of this article, and any mention of color refers to that version